difference
Adam 2479
3702
Betty 6181
2468
Dan 8649
The GCD of the differences is 1234, which is Jerry's number. All the remainders are 11.
Bonus:
Of course it's assumed Jerry is doing such a GCD calculation. The only question is How impressive is his calculated number. I don't know if it's implied that Jerry's number must be 4 digits. If that's the question then the probability is 1 - 0.99933419926 (see below); that's about 1 in 1502. In a simulation of 10,000 trials it happened only twice (see also below).
First we need a formula for the number of ways of getting a given pair of differences.
First of all it does not matter whether the order of any given set of three numbers a, b and c is abc, acb, bac, bca, cab or cba. Each is equally likely, with the caveat that aab is one of only three, not six, and aaa has no permutations.
We know that only 9000 out of the 9000^3 possible choices made by A, B and D have all three digits the same, so both differences are zero. (Yes here we're considering all orders separately, but we're just calculating the ratio--the probability that both differences are zero.) The probability that both differences are zero is 1/9000^2 ~= 0.00000001234.... In these cases no audience will be impressed that Jerry repeats back the number given, or any other number for that matter.
Similarly the probability that two of the numbers match and one is different is (9000*8999*3)/(9000^3) = 3*8999/9000^2 ~= 0.00033296.... The best Jerry can do in this instance is to give the GCD of the two numbers and the remainder is zero in both cases.
9000^3 = 729,000,000,000
Those with one or two matching numbers is 9000+9000*8999*3 = 242,982,000, leaving 728,757,018,000 where we're concerned with the GCD of two non-zero numbers.
By the below table, the overall probability that there are three distinct numbers chosen and that the GCD of the three is 1 is about 60.8%. So it's less than 50% likely that Jerry can come up with any number at all which would act as such a divisor. Then there's that approximately 15% probability that there would be such a divisor, but it would be 2--hardly amazing. In fact, the probability is less than 0.0007 (i.e., less than 7/100 of 1%) that the GCD would even be a 4-digit number.
Propabilities out of all 729000000000 possible configurations:
(while the denominator for the probability is
729000000000, the cumulative probability does not
include the above probabilities of any matching numbers.)
GCD number cumulative cumulative
number probability
1 443178686352 443178686352 0.60792686742
2 110794710792 553973397144 0.75990863806
3 49242120084 603215517228 0.82745612788
4 27698603232 630914120460 0.86545146840
5 17727171780 648641292240 0.88976857646
6 12310427448 660951719688 0.90665530821
7 9044391612 669996111300 0.91906188107
8 6924651696 676920762996 0.92856071742
9 5471295228 682392058224 0.93606592349
10 4431773520 686823831744 0.94214517386
11 3662582244 690486413988 0.94716929216
12 3077597232 693564011220 0.95139096189
13 2622335724 696186346944 0.95498813024
14 2261038272 698447385216 0.95808969165
15 1969595100 700416980316 0.96079146820
16 1731112752 702148093068 0.96316610846
17 1533483480 703681576548 0.96526965233
18 1367801208 705049377756 0.96714592285
19 1227619692 706276997448 0.96882990048
20 1107880800 707384878248 0.97034962723
21 1004901696 708389779944 0.97172809320
22 915611640 709305391584 0.97298407625
23 837712044 710143103628 0.97413320114
24 769357584 710912461212 0.97518856133
25 709023300 711621484512 0.97616115845
26 655532664 712277017176 0.97706038021
27 607891824 712884909000 0.97789425103
28 565267008 713450176008 0.97866965159
29 526925760 713977101768 0.97939245784
30 492404760 714469506528 0.98006791019
31 461134548 714930641076 0.98070046787
32 432729168 715363370244 0.98129406069
33 406904472 715770274716 0.98185222869
34 383313192 716153587908 0.98237803554
35 361717680 716515305588 0.98287421891
36 341909856 716857215444 0.98334323106
37 323673948 717180889392 0.98378722825
38 306849816 717487739208 0.98420814706
39 291320964 717779060172 0.98460776430
40 276936720 718055996892 0.98498765006
41 263616084 718319612976 0.98534926334
42 251213832 718570826808 0.98569386393
43 239664864 718810491672 0.98602262232
44 228897648 719039389320 0.98633661086
45 218814480 719258203800 0.98663676790
46 209398632 719467602432 0.98692400882
47 200572140 719668174572 0.98719914207
48 192315024 719860489596 0.98746294869
49 184542840 720045032436 0.98771609388
50 177217200 720222249636 0.98795919017
51 170341812 720392591448 0.98819285521
52 163838112 720556429560 0.98841759885
53 157719804 720714149364 0.98863394974
54 151921872 720866071236 0.98884234737
55 146449800 721012521036 0.98904323873
56 141268176 721153789212 0.98923702224
57 136352772 721290141984 0.98942406308
58 131696808 721421838792 0.98960471714
59 127270596 721549109388 0.98977929957
60 123054480 721672163868 0.98994809858
61 119061216 721791225084 0.99011141987
62 115252992 721906478076 0.99026951725
63 111619944 722018098020 0.99042263103
64 108161040 722126259060 0.99057100008
65 104847180 722231106240 0.99071482337
66 101693736 722332799976 0.99085432095
67 98688180 722431488156 0.99098969569
68 95803584 722527291740 0.99112111350
69 93041712 722620333452 0.99124874273
70 90401280 722710734732 0.99137274998
71 87866028 722798600760 0.99149327951
72 85444848 722884045608 0.99161048780
73 83129052 722967174660 0.99172451942
74 80898216 723048072876 0.99183549091
75 78760800 723126833676 0.99194353042
76 76702128 723203535804 0.99204874596
77 74720472 723278256276 0.99215124318
78 72812736 723351069012 0.99225112347
79 70974024 723422043036 0.99234848153
80 69201360 723491244396 0.99244340795
81 67498488 723558742884 0.99253599847
82 65856000 723624598884 0.99262633592
83 64269456 723688868340 0.99271449704
84 62738640 723751606980 0.99280055827
85 61279140 723812886120 0.99288461745
86 59865864 723872751984 0.99296673798
87 58488012 723931239996 0.99304696844
88 57165456 723988405452 0.99312538471
89 55881744 724044287196 0.99320204005
90 54656640 724098943836 0.99327701486
91 53465376 724152409212 0.99335035557
92 52310544 724204719756 0.99342211215
93 51188472 724255908228 0.99349232953
94 50114784 724306023012 0.99356107409
95 49065720 724355088732 0.99362837960
96 48053520 724403142252 0.99369429664
97 47068200 724450210452 0.99375886207
98 46113024 724496323476 0.99382211725
99 45181800 724541505276 0.99388409503
100 44286000 724585791276 0.99394484400
...
984 418896 728508474540 0.99932575383
985 417840 728508892380 0.99932632700
986 416784 728509309164 0.99932689872
987 415728 728509724892 0.99932746899
988 414672 728510139564 0.99932803781
989 413616 728510553180 0.99932860519
990 412560 728510965740 0.99932917111
991 411504 728511377244 0.99932973559
992 410448 728511787692 0.99933029862
993 409392 728512197084 0.99933086020
994 408336 728512605420 0.99933142033
995 407280 728513012700 0.99933197901
996 406224 728513418924 0.99933253625
997 405168 728513824092 0.99933309203
998 404112 728514228204 0.99933364637
999 403056 728514631260 0.99933419926
1000 402000 728515033260 0.99933475070
728757018000
729000000000 done
A simulation with 10000 trials verifies the general nature of these probabilities:
(The numbers add up to 9999--there was one case of a match among the 3 numbers drawn)
GCD Number of occurrences
1 6098
2 1556
3 629
4 380
5 240
6 180
7 121
8 89
9 84
10 44
11 51
12 42
13 35
14 35
15 21
16 21
17 21
18 18
19 22
20 14
21 22
22 11
23 16
24 5
25 14
26 7
27 5
28 11
29 5
30 2
31 5
32 8
33 6
34 4
35 3
36 5
37 2
38 3
39 3
40 3
41 1
42 5
43 4
44 5
45 4
46 5
47 1
48 2
49 4
50 1
51 1
52 3
54 2
55 3
56 7
57 2
58 2
59 1
60 1
61 2
62 4
63 2
64 3
66 1
67 2
69 1
70 1
71 1
72 1
73 3
75 1
76 2
77 4
78 2
79 1
80 1
81 1
82 1
83 1
84 2
85 1
86 3
87 1
88 1
90 2
91 1
92 1
94 2
96 1
97 1
98 2
99 2
102 1
104 1
105 2
107 1
109 1
114 1
119 1
128 1
134 1
138 2
140 2
152 2
156 2
159 2
171 1
180 2
185 1
195 1
217 1
224 1
230 1
251 1
274 1
287 2
291 1
302 1
329 1
347 1
390 1
425 1
429 1
452 1
472 1
492 1
521 1
543 1
878 1
964 1
969 1
1189 1
3856 1
For tr = 1 To 10000
a = 1000 + Int(Rnd(1) * 10000)
b = 1000 + Int(Rnd(1) * 10000)
c = 1000 + Int(Rnd(1) * 10000)
If a <> b And b <> c And a <> c Then
If b < a Then h = b: b = a: a = h
If c < a Then h = c: c = a: a = h
If c < b Then h = b: b = c: c = h
d1 = b - a: d2 = c - b
g(gcd(d1, d2)) = g(gcd(d1, d2)) + 1
If tr Mod 100 = 0 Then
Text1.Text = Text1.Text & tr
End If
End If
DoEvents
Next
Text1.Text = Text1.Text & crlf
For i = 1 To 10000
If g(i) > 0 Then
Text1.Text = Text1.Text & i & " " & g(i) & crlf
End If
DoEvents
Next
...
Function gcd(a, b)
x = a: y = b
Do
q = Int(x / y)
z = x - q * y
x = y: y = z
Loop Until z = 0
gcd = x
End Function
proposed solution
